Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{y^2 - 4}{y + 2}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = y$ $ b = \sqrt{4} = 2$ So we can rewrite the expression as: $q = \dfrac{({y} + {2})({y} {-2})} {y + 2} $ We can divide the numerator and denominator by $(y + 2)$ on condition that $y \neq -2$ Therefore $q = y - 2; y \neq -2$